Preface -- 1 Introduction -- 1.1 Gyroscopic stabilization on a rotating surface -- 1.1.1 Brouwer's mechanical model -- 1.1.2 Eigenvalue problems and the characteristic equation -- 1.1.3 Eigencurves and bifurcation of multiple eigenvalues -- 1.1.4 Singular stability boundary of the rotating saddle trap -- 1.2 Manifestations of Brouwer's model in physics -- 1.2.1 Stability of deformable rotors -- 1.2.2 Foucault's pendulum, Bryan's effect, Coriolis vibratory gyroscopes, and the Hannay-Berry phase -- 1.2.3 Polarized light within a cholesteric liquid crystal -- 1.2.4 Helical magnetic quadrupole focussing systems -- 1.2.5 Modulational instability -- 1.3 Brouwer's problem with damping and circulatory forces -- 1.3.1 Circulatory forces -- 1.3.2 Dissipation-induced instability of negative energy modes -- 1.3.3 Circulatory systems and the destabilization paradox -- 1.3.4 Merkin's theorem, Nicolai's paradox, and subcritical flutter -- 1.3.5 Indefinite damping and parity-time (PT) symmetry -- 1.4 Scope of the book -- 2 Lyapunov stability and linear stability analysis -- 2.1 Main facts and definitions -- 2.1.1 Stability, instability, and uniform stability -- 2.1.2 Attractivity and asymptotic stability -- 2.1.3 Autonomous, nonautonomous, and periodic systems -- 2.2 The direct (second) method of Lyapunov -- 2.2.1 Lyapunov functions -- 2.2.2 Lyapunov and Persidskii theorems on stability -- 2.2.3 Chetaev and Lyapunov theorems on instability -- 2.3 The indirect (first) method of Lyapunov -- 2.3.1 Linearization -- 2.3.2 The characteristic exponent of a solution -- 2.3.3 Lyapunov regularity of linearization -- 2.3.4 Stability and instability in the first approximation -- 2.4 Linear stability analysis -- 2.4.1 Autonomous systems -- 2.4.2 Lyapunov transformation and reducibility -- 2.4.3 Periodic systems.

2.4.4 Example. Coupled parametric oscillators -- 2.5 Algebraic criteria for asymptotic stability -- 2.5.1 Lyapunov's matrix equation and stability criterion -- 2.5.2 The Leverrier-Faddeev algorithm and Lewin's formula -- 2.5.3 Müller's solution to the matrix Lyapunov equation -- 2.5.4 Inertia theorems and observability index -- 2.5.5 Hermite's criterion via the matrix Lyapunov equation -- 2.5.6 Routh-Hurwitz, Liénard-Chipart, and Bilharz criteria -- 2.6 Robust Hurwitz stability vs. structural instability -- 2.6.1 Multiple eigenvalues: singularities and structural instabilities -- 2.6.2 Multiple eigenvalues: spectral abscissa minimization and robust stability -- 3 Hamiltonian and gyroscopic systems -- 3.1 Sobolev's top and an indefinite metric -- 3.2 Elements of Pontryagin and Krein space theory -- 3.3 Canonical and Hamiltonian equations -- 3.3.1 Krein signature of eigenvalues -- 3.3.2 Krein collision or linear Hamiltonian-Hopf bifurcation -- 3.3.3 MacKay's cones, veering, and instability bubbles -- 3.3.4 Instability degree and count of eigenvalues -- 3.3.5 Graphical interpretation of the Krein signature -- 3.3.6 Strong stability: robustness to Hamiltonian's variation -- 3.3.7 Inertia theorems and stability of gyroscopic systems -- 3.3.8 Positive and negative energy modes and Krein signature -- 3.3.9 Dispersive wave propagation in conservative systems -- 3.3.10 Absolute and convective instability -- 4 Reversible and circulatory systems -- 4.1 Reversible systems -- 4.2 Nonconservative positional forces -- 4.3 Circulatory systems -- 4.3.1 Divergence and flutter instabilities -- 4.3.2 Multiple parameter families of circulatory systems -- 4.3.3 Generic singularities on the stability boundary -- 4.4 Perturbation of eigenvalues -- 4.4.1 Simple eigenvalue.

4.4.2 Double eigenvalue of geometric multiplicity 1 -- 4.4.3 Double eigenvalue of geometric multiplicity 2 -- 4.4.4 Triple eigenvalue of geometric multiplicity 1 -- 4.5 Geometry of the stability boundary -- 4.5.1 Linear and quadratic approximations at smooth points -- 4.5.2 Singularities in two-parameter circulatory systems -- 4.5.3 Example. Stabilization of comfortable walking -- 4.5.4 Singularities in three-parameter circulatory systems -- 4.5.5 The cone aa and Merkin's instability theorem -- 4.5.6 Example: a brake disk in distributed frictional contact -- 4.5.7 Example: stability of an airfoil in an inviscid flow -- 4.6 Eigencurves, their crossing and veering -- 4.6.1 Convex flutter domain: conical point aa -- 4.6.2 Convex/concave flutter domain: smooth points a2 -- 4.7 Parametric optimization of circulatory systems -- 4.7.1 Example: optimization of Ziegler's pendulum -- 4.7.2 A nonsmooth and nonconvex optimization problem -- 4.7.3 The gradient of the critical load -- 4.7.4 An infinite gradient at the crossing of the eigencurves -- 4.7.5 Improving variations and necessary conditions for optimality in the case where the eigencurves cross -- 5 Influence of structure of forces on stability -- 5.1 Undamped potential systems -- 5.1.1 Lagrange's theorem and Poincaré instability degree -- 5.1.2 Rayleigh's theorem on movement of eigenvalues -- 5.1.3 Steady-state bifurcation -- 5.2 Damped potential systems -- 5.2.1 Overdamped and heavily damped systems -- 5.2.2 Indefinitely damped systems -- 5.3 Undamped gyroscopic systems -- 5.3.1 Extension of Rayleigh's theorem -- 5.3.2 Criteria of gyroscopic stabilization -- 5.4 Damped gyroscopic systems -- 5.4.1 Kelvin-Tait-Chetaev theorem -- 5.5 Circulatory systems with and without velocity-dependent forces.

5.5.1 Merkin's theorem and Bulatovic's flutter condition -- 5.5.2 Bottema-Lakhadanov-Karapetyan theorem -- 5.5.3 Stabilizing and destabilizing damping configurations -- 6 Dissipation-induced instabilities -- 6.1 Crandall's gyropendulum -- 6.1.1 Conservative gyroscopic stabilization and its destruction by stationary damping -- 6.1.2 Singular threshold of the nonconservative gyroscopic stabilization -- 6.1.3 Imperfect Krein collision and exchange of instability between negative and positive energy modes -- 6.2 Gyroscopic stabilization of nonconservative systems -- 6.2.1 The case of m = 2 degrees of freedom -- 6.2.2 The case of arbitrary even m -- 6.3 Near-Hamiltonian systems -- 6.4 Gyroscopic and circulatory systems as limits of dissipative systems -- 7 Nonself-adjoint boundary eigenvalue problems -- 7.1 Adjoint boundary eigenvalue problems -- 7.2 Perturbation of eigenvalues -- 7.2.1 Semisimple eigenvalues -- 7.2.2 Multiple eigenvalues with the Keldysh chain -- 7.2.3 Higher order perturbation terms for double nonderogatory eigenvalues -- 7.2.4 Degenerate splitting of double nonderogatory eigenvalues -- 7.3 Example: a rotating circular string with an elastic restraint -- 7.4 Example: the Herrmann-Smith paradox -- 7.4.1 Formulation of the problem -- 7.4.2 Stationary flutter domain and mobile divergence region -- 7.4.3 Sensitivity of the critical flutter load to the redistribution of the elasticity modulus -- 7.5 Example: Beck's column loaded by a partially follower force -- 7.5.1 The stability-divergence boundary (point A) -- 7.5.2 The flutter threshold of Beck's column (point C) -- 7.5.3 The singularity O2 on the stability boundary (point B) -- 8 Destabilization paradox in continuous circulatory systems -- 8.1 Movement of eigenvalues under a velocity-dependent perturbation.

8.1.1 Generalized boundary eigenvalue problem -- 8.1.2 Variation of parameters that is transversal to the stability boundary -- 8.1.3 Variation of parameters that is tangential to the stability boundary -- 8.1.4 Transfer of instability between modes -- 8.1.5 Drop in the critical frequency -- 8.2 Singular threshold of the flutter instability -- 8.2.1 Drop in the critical flutter load -- 8.2.2 The "no drop" condition and the tangent cone to the domain of asymptotic stability -- 8.3 Example: dissipation-induced instability of Beck's column -- 8.3.1 Beck's column without damping -- 8.3.2 Beck's column with Kelvin-Voigt and viscous damping -- 8.3.3 Viscoelastic Beck's column with a dash-pot -- 8.3.4 Ziegler's pendulum with a dash-pot -- 8.4 Application to finite-dimensional systems -- 8.4.1 The destabilization paradox in Ziegler's pendulum -- 9 MHD kinematic mean field α2-dynamo -- 9.1 Eigenvalue problem for α2-dynamo -- 9.2 Uniform α-profiles generate only nonoscillatory dynamos -- 9.2.1 Conducting exterior: self-adjointness in a Krein space -- 9.2.2 Basis properties of eigenfunctions -- 9.2.3 Spectral mesh of eigencurves -- 9.2.4 Deformation of the spectral mesh via transition from conducting to insulating surrounding -- 9.3 Nonhomogeneous a-profiles and the conducting exterior -- 9.3.1 I = 0: definite Krein signature prohibits formation of complex eigenvalues -- 9.3.2 l = 0: oscillating solutions from the repeated decaying modes with mixed Krein signature -- 9.3.3 l = 0: Fourier components of a(x) determine the unfolding pattern of the spectral mesh -- 9.4 Insulating boundary conditions induce unstable oscillations -- 9.4.1 l = 0: complex unfolding of double eigenvalues with definite Krein signature -- 10 Campbell diagrams and subcritical friction-induced flutter.

10.1 Friction-induced vibrations and sound generation.